Question

Suppose \(T: V \rightarrow V\) is a linear operator with the property that \(T[T(\mathbf{v})]=\mathbf{v}\) for all \(\mathbf{v}\) in \(V\). (For example, transposition in \(\mathbf{M}_{n n}\) or conjugation in \(\mathbb{C} .\) ) If \(\mathbf{v} \neq \mathbf{0}\) in \(V\), show that \(\\{\mathbf{v}, T(\mathbf{v})\\}\) is linearly independent if and only if \(T(\mathbf{v}) \neq \mathbf{v}\) and \(T(\mathbf{v}) \neq-\mathbf{v}\).

Short answer

\(\{\mathbf{v}, T(\mathbf{v})\}\) is linearly independent if and only if \(T(\mathbf{v}) \neq \mathbf{v}\) and \(T(\mathbf{v}) \neq -\mathbf{v}\).

Step-by-step solution

Establish Linear Independence Definition

A set of vectors \{\mathbf{v}_1, \mathbf{v}_2\} is linearly independent if and only if the equation \(a\mathbf{v}_1 + b\mathbf{v}_2 = \mathbf{0}\) implies that \(a = 0\) and \(b = 0\). We will use this definition for the set \{\mathbf{v}, T(\mathbf{v})\}.

Assume Linear Independence

Assume that \{\mathbf{v}, T(\mathbf{v})\} is linearly independent. This implies that if \(a\mathbf{v} + bT(\mathbf{v}) = \mathbf{0}\), then \(a = 0\) and \(b = 0\). We need to demonstrate that \(T(\mathbf{v}) eq \mathbf{v}\) and \(T(\mathbf{v}) eq -\mathbf{v}\).

Proof by Contradiction for T(v) = v

Assume \(T(\mathbf{v}) = \mathbf{v}\). Then our operator becomes \(T[T(\mathbf{v})] = T(\mathbf{v}) = \mathbf{v}\), which means \(\mathbf{v} = \mathbf{v}\). However, if \(T(\mathbf{v}) = \mathbf{v}\), this would imply that the vectors \(\mathbf{v}\) and \(T(\mathbf{v}) = \mathbf{v}\) are not linearly independent. This contradicts our original assumption of independence.

Proof by Contradiction for T(v) = -v

Assume \(T(\mathbf{v}) = -\mathbf{v}\). Then \(T[T(\mathbf{v})] = T(-\mathbf{v}) = -T(\mathbf{v}) = -(-\mathbf{v}) = \mathbf{v}\) holds as expected. However, if \(T(\mathbf{v}) = -\mathbf{v}\), the vectors \(\mathbf{v}\) and \(T(\mathbf{v}) = -\mathbf{v}\) would not be linearly independent since \(\mathbf{v} + T(\mathbf{v}) = \mathbf{v} - \mathbf{v} = \mathbf{0}\), contradicting our assumption.

Conclusion of "if" Implication

Therefore, if \{\mathbf{v}, T(\mathbf{v}) \} is linearly independent, then \(T(\mathbf{v}) eq \mathbf{v}\) and \(T(\mathbf{v}) eq -\mathbf{v}\).

Assume the "only if" Condition

Assume \(T(\mathbf{v}) eq \mathbf{v}\) and \(T(\mathbf{v}) eq -\mathbf{v}\). We need to prove that \{\mathbf{v}, T(\mathbf{v})\} is linearly independent.

Confirms Linear Independence with Assumptions

Given \(a\mathbf{v} + bT(\mathbf{v}) = \mathbf{0}\), we apply \(T\) to the equation: \(aT(\mathbf{v}) + bT[T(\mathbf{v})] = \mathbf{0}\), which implies \(aT(\mathbf{v}) + b\mathbf{v} = \mathbf{0}\). This yields two equations \(a + b = 0\) and \(b + a = 0\), concluding \(a = 0\) and \(b = 0\) since \(T(\mathbf{v})\) is neither \(\mathbf{v}\) nor \(-\mathbf{v}\), thus confirming linear independence.

Key concepts

Linear Operators

A linear operator is a crucial concept in linear algebra, often described as a function that maps vectors from one space to another, while preserving vector addition and scalar multiplication. In simpler terms, linear operators act on vectors in a consistent, predictable way, ensuring that they remain within the same vector space (in this case, from and back to \(V\)).

• **Linearity**: For a set of vectors and scalars, \(T(a\boldsymbol{v} + b\boldsymbol{w}) = aT(\boldsymbol{v}) + bT(\boldsymbol{w})\) where \(a\) and \(b\) are scalars.
• **Identity Action**: The given property \(T[T(\boldsymbol{v})] = \boldsymbol{v}\) implies a special type of consistency that makes linear operators very useful in practical scenarios, like transformations.

Linear operators appear widely, from simple matrix transformations to complex functions in quantum mechanics. Understanding their properties is vital for solving many algebraic problems by manipulating the structure of vector spaces.

Vector Spaces

Vector spaces form the foundation of understanding linear algebra as they provide the backdrop where linear operators and independence concepts exist. A vector space is essentially a collection of vectors, where addition and multiplication by numbers (scalars) are defined and obey certain rules.
Imagine a vector space like a habitat, allowing vectors to interact freely under specified rules. These rules include:

• **Closure under addition and scalar multiplication**: Any two vectors added together, or any vector multiplied by a scalar, must remain in the vector space.
• **Existence of a zero vector**: There must be a vector that, when added to any vector, leaves it unchanged.

In the problem scenario, the space \(V\) holds vectors \(\boldsymbol{v}\) and \(T(\boldsymbol{v})\), and the set \{\boldsymbol{v}, T(\boldsymbol{v})\} needs to maintain independence—a key property allowing unique vector configurations, crucial for multiple applications in engineering, physics, and beyond.

Proof by Contradiction

Proof by contradiction is a logical method used to establish the truth of a statement by demonstrating that assuming the contrary leads to an inconsistency. In more straightforward terms, if assuming the opposite of what you want to prove results in something impossible or illogical, then the original statement must be true.
In the exercise, to verify the linear independence condition, we assume what we wish to disprove: whether \(T(\boldsymbol{v}) = \boldsymbol{v}\) or \(T(\boldsymbol{v}) = -\boldsymbol{v}\) makes the vectors \(\{\boldsymbol{v}, T(\boldsymbol{v})\}\) dependent. Both assumptions result in a contradiction:

• **Assumption **\(T(\boldsymbol{v}) = \boldsymbol{v}\): Leads to \(\boldsymbol{v} = T T(\boldsymbol{v}) = T(\boldsymbol{v})\), meaning \(\boldsymbol{v}\) and \(T(\boldsymbol{v}) \) would be the same vector, hence they are dependent.
• **Assumption **\(T(\boldsymbol{v}) = -\boldsymbol{v}\): Results in \(\boldsymbol{v} + T(\boldsymbol{v}) = 0\), implying dependence as their linear combination gives the zero vector.

Therefore, by ruling out these possibilities, we confirm that for \(\{\boldsymbol{v}, T(\boldsymbol{v})\}\) to be independent, neither \(T(\boldsymbol{v})\) equal to \(\boldsymbol{v}\) nor \(-\boldsymbol{v}\) should hold. This logical technique is prevalent in mathematics to simplify complex proofs.